A piece of cloth cost ₹ 120. If the piece, then was 2 m longer and each metre of cloth costs ₹ 2 less than cost of the piece of cloth would have remained unchanged. What is the original rate per metre of the cloth?

Let the length of the piece be L m.
Then,rate per metre = ₹ 120/L
New length = (L + 2) m
Since, the cost remains same.
∴ New rate per metre =Rs.120/(L + 2)
According to the given condition, 120/L - 120/(L + 2) = 2
⇒ [120(L + 2) - 120L]/L(L + 2) = 2
⇒ [120L + 240 - 120L] / L(L + 2) = 2
⇒ [120L + 240 -120L] / L(L + 2) = 2
⇒ 240/L(L + 2) = 2
⇒ 240 = 2L(L + 2)
⇒ 120 = L(L + 2) = L² + 2L
⇒ L² + 2L - 120 = 0
⇒ L² + 12L - 10L - 120 = 0
⇒ L(L + 12) - 10 x (L + 12) = 0
⇒ (L + 12)(L - 10) = 0
⇒ L + 12 = 0 or L - 10 = 0
⇒ L = -12 or 10
⇒ L = 10(since, L cannot be negative)
∴ Original rate per metre of cloth = 120/L = 120/10 = ₹ 12